Finite Group Databasebeta

← C2×#20273666 #27356161

Subgroups — C2×#20273666 #27356161

Order 13568.

1652 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1652 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811185C211yesmaximal, characteristic
2C2×#16810762C212yesmaximal, complemented
3#20273666C218yesmaximal, direct product factor
4C2×#16810356C212yesmaximal, complemented
5C2×#16810347C211yesmaximal, characteristic, complemented
6C2×C53×#393C211yesmaximal, characteristic, complemented
7C23×C22⋊D53—21no—
8C22×#11719924—22no—
9#16811185—24no—
10#16811185—22no—
11#16811185C2214yes—
12C2×#13133201—22no—
13C2×#13133191—21no—
14C2×#13133191C2212yes—
15C2×#13133187—22no—
16#16810762—24no—
17#16810762C2218yescomplemented
18C2×#13133141—22no—
19C2×C22⋊#88322C2212yescomplemented
20#16810356C2218yescomplemented
21#16810347C2214yescomplemented
22C23×Q8×C53C2211yescharacteristic, complemented
23C22×C22⋊C4×C53—21no—
24C22×C22⋊C4×C53C2211yescharacteristic
25C53×#393—22no—
26C53×#393C2214yescomplemented
27C2×#95×C53—21no—
28C2×#95×C53C2211yescharacteristic, complemented
29C24×D53—21no—
30C22×D212—21no—
31C22×D212D412yes—
32C22×C22⋊D53—22no—
33C22×C22⋊D53—24no—
34C22×C22⋊D53—22no—
35C22×C22⋊D53—24no—
36C22×C22⋊D53—44no—
37C22×C22⋊D53—44no—
38C23×Dic53—21no—
39C2×#11719924—28no—
40C2×#11719924—48no—
41C22×#88322—22no—
42#13133201—28no—
43#13133201—28no—
44#13133191—24no—
45#13133191—24no—
46#13133191—22no—
47#13133191—22no—
48#13133191D414yes—
49#13133191C2314yes—
50#13133187—28no—
51#13133187—28no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53D412yes—
54C2×#11719921—22no—
55C2×#11719921—22no—
56#13133141—28no—
57C22⋊#88322—22no—
58C22⋊#88322C2314yescomplemented
59C25×C53D411yescharacteristic
60C22×Q8×C53—21no—
61C22×Q8×C53—22no—
62C22×Q8×C53—22no—
63C22×Q8×C53—44no—
64C22×Q8×C53C2312yescomplemented
65C23×C4×C53—21no—
66C23×C4×C53—21no—
67C23×C4×C53D411yescharacteristic
68C23×C4×C53C2311yescharacteristic
69C2×C22⋊C4×C53—24no—
70C2×C22⋊C4×C53—22no—
71C2×C22⋊C4×C53—22no—
72C2×C22⋊C4×C53—44no—
73C2×C22⋊C4×C53D412yes—
74C2×C22⋊C4×C53C2312yes—
75#95×C53—24no—
76#95×C53—21no—
77#95×C53C2312yescomplemented
78C24×C53—21no—
79C24×C53—21no—
80C24×C53—22no—
81C24×C53—42no—
82C24×C53—42no—
83C24×C53SD1614yescomplemented
84C24×C53C2×D411yescharacteristic
85C24×C53C2×D412yes—
86C23×D53—22no—
87C23×D53—24no—
88C23×D53—44no—
89C23×D53—82no—
90C23×D53—82no—
91C2×Q8×C53—28no—
92C2×Q8×C53—24no—
93C2×Q8×C53—44no—
94C2×Q8×C53—44no—
95C2×Q8×C53—44no—
96C2×Q8×C53—48no—
97C2×Q8×C53—42no—
98C22×C4×C53—24no—
99C22×C4×C53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—21no—
102C22×C4×C53—22no—
103C22×C4×C53—42no—
104C22×C4×C53—42no—
105C22×C4×C53—41no—
106C22×C4×C53—41no—
107C22×C4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53—42no—
110C22×C4×C53—42no—
111C22×C4×C53C2×D412yes—
112C22×C4×C53C2×D412yes—
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D412yes—
115C22×C4×C53C2411yescharacteristic
116C2×C22⋊D53—28no—
117C2×C22⋊D53—24no—
118C2×C22⋊D53—24no—
119C2×C22⋊D53—48no—
120C2×C22⋊D53—416no—
121C2×C22⋊D53—48no—
122C2×C22⋊D53—44no—
123C2×C22⋊D53—48no—
124C2×C22⋊D53—44no—
125C22×Dic53—24no—
126C22×Dic53—22no—
127C22×Dic53—44no—
128C2×D212—24no—
129C2×D212—22no—
130C2×D212—44no—
131C2×D212—82no—
132C2×D212C2×D414yes—
133C22⋊C4×C53—24no—
134C22⋊C4×C53—24no—
135C22⋊C4×C53—48no—
136#11719924—416no—
137#11719921—28no—
138#11719921—24no—
139#11719921—24no—
140C4⋊Dic53—24no—
141C4⋊Dic53—22no—
142C4⋊Dic53C2×D414yes—
143C2×#88322—24no—
144C2×#88322—44no—
145C23×C53—22no—
146C23×C53—22no—
147C23×C53—22no—
148C23×C53—24no—
149C23×C53—22no—
150C23×C53—24no—
151C23×C53—48no—
152C23×C53—42no—
153C23×C53—42no—
154C23×C53—42no—
155C23×C53—42no—
156C23×C53—84no—
157C23×C53—81no—
158C23×C53—81no—
159C23×C53#12011yescharacteristic
160C23×C53#12011yescharacteristic
161C23×C53C2×SD1612yes—
162C23×C53C2×SD1614yescomplemented
163C23×C53AGL(1,ℤ/8)12yes—
164C23×C53C22×D411yescharacteristic
165C22×D53—24no—
166C22×D53—48no—
167C22×D53—44no—
168C22×D53—44no—
169C22×D53—44no—
170C22×D53—84no—
171C22×D53—84no—
172C22×D53—816no—
173Q8×C53—28no—
174Q8×C53—44no—
175Q8×C53—48no—
176C2×C4×C53—24no—
177C2×C4×C53—24no—
178C2×C4×C53—22no—
179C2×C4×C53—42no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—42no—
183C2×C4×C53—44no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—44no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—81no—
192C2×C4×C53—84no—
193C2×C4×C53#12012yes—
194C2×C4×C53C22×D412yes—
195C22⋊D53—416no—
196C22⋊D53—44no—
197C22⋊D53—44no—
198C22⋊D53—48no—
199C2×Dic53—24no—
200C2×Dic53—48no—
201C2×Dic53—44no—
202D212—44no—
203D212—82no—
204D212—84no—
205#88322—44no—
206C2×#1601—531nomaximal, Sylow
207C22×C53—24no—
208C22×C53—24no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—24no—
214C22×C53—44no—
215C22×C53—48no—
216C22×C53—42no—
217C22×C53—42no—
218C22×C53—82no—
219C22×C53—84no—
220C22×C53#45012yes—
221C22×C53#45012yes—
222C22×C53#45512yes—
223C22×C53B(4,𝔽2)12yes—
224C22×C53C2×#12011yescharacteristic
225C22×C53C22×SD1611yescharacteristic
226C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
227C2×D53—44no—
228C2×D53—44no—
229C2×D53—44no—
230C2×D53—816no—
231C2×D53—88no—
232C4×C53—22no—
233C4×C53—22no—
234C4×C53—42no—
235C4×C53—44no—
236C4×C53—82no—
237Dic53—44no—
238C2×#534—531no—
239C2×#448—532no—
240C2×#393D5311yescharacteristic, complemented
241#1601—538no—
242C2×#327—532no—
243C2×#323—531no—
244C2×C53—21no—
245C2×C53—21no—
246C2×C53—22no—
247C2×C53—44no—
248C2×C53#160114yescomplemented
249C2×C53C2×#45011yescharacteristic
250C2×C53C2×#45511yescharacteristic
251C2×C53C2×B(4,𝔽2)11yescharacteristic
252D53—84no—
253C23×Q8C2×D5311yescharacteristic, complemented
254C23×D4—1061no—
255C22×SD16—1062no—
256#534—534no—
257#534—1064no—
258#534—1062no—
259C2×#123—1062no—
260C2×#121—532no—
261C2×#121—1061no—
262C2×#120—1062no—
263C22×C22⋊C4—21no—
264C22×C22⋊C4C2×D5311yescharacteristic
265#448—538no—
266#448—1064no—
267C2×#103—1062no—
268C2×C22⋊C8—532no—
269#393—22no—
270#393C2×D5314yescomplemented
271C2×#95—21no—
272C2×#95C2×D5311yescharacteristic, complemented
273#327—538no—
274#323—534no—
275C53C2×#160111yescharacteristic, complemented, Sylow
276C25—1061no—
277C25C22⋊D5311yescharacteristic
278C22×Q8—21no—
279C22×Q8—22no—
280C22×Q8—22no—
281C22×Q8—44no—
282C22×Q8C22×D5312yescomplemented
283C22×D4—532no—
284C22×D4—1061no—
285C22×D4—1062no—
286C22×D4—1064no—
287C22×D4—1062no—
288C22×D4—1064no—
289C22×D4—2124no—
290C22×D4—2124no—
291C23×C4—21no—
292C23×C4—21no—
293C23×C4—1061no—
294C23×C4C22⋊D5311yescharacteristic
295C23×C4C22×D5311yescharacteristic
296C2×SD16—1068no—
297C2×SD16—2128no—
298C22×C8—1062no—
299#123—1068no—
300#123—1068no—
301#121—534no—
302#121—534no—
303#121—1064no—
304#121—1062no—
305#121—1062no—
306#121—1064no—
307#120—1068no—
308#120—1068no—
309C2×C4⋊C4—532no—
310C2×C4⋊C4—1061no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—1062no—
316C2×C22⋊C4—1062no—
317C2×C22⋊C4C22⋊D5312yes—
318C2×C22⋊C4C22×D5312yes—
319#103—1068no—
320C22⋊C8—534no—
321C22⋊C8—1062no—
322#95—24no—
323#95—21no—
324#95C22×D5312yescomplemented
325C24—21no—
326C24—21no—
327C24—22no—
328C24—42no—
329C24—42no—
330C24—1064no—
331C24—1062no—
332C24—4242no—
333C24—4242no—
334C24—2124no—
335C24#1171992414yescomplemented
336C24C2×C22⋊D5311yescharacteristic
337C24C2×C22⋊D5312yes—
338C2×Q8—28no—
339C2×Q8—24no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—48no—
344C2×Q8—42no—
345C2×D4—534no—
346C2×D4—1064no—
347C2×D4—1068no—
348C2×D4—1064no—
349C2×D4—1064no—
350C2×D4—1062no—
351C2×D4—4242no—
352C2×D4—2124no—
353C2×D4—21216no—
354C2×D4—2128no—
355C2×D4—2128no—
356C2×D4—2124no—
357C2×D4—2128no—
358C2×D4—2124no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—1064no—
373C22×C4—1062no—
374C22×C4—2124no—
375C22×C4D4×D5312yes—
376C22×C4D4×D5312yes—
377C22×C4C2×C22⋊D5312yes—
378C22×C4C2×C22⋊D5311yescharacteristic
379C22×C4C23×D5311yescharacteristic
380SD16—21216no—
381C2×C8—1064no—
382C2×C8—2124no—
383C4⋊C4—534no—
384C4⋊C4—1064no—
385C4⋊C4—1062no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—48no—
389C22⋊C4—1068no—
390C22⋊C4—1064no—
391C22⋊C4—1064no—
392C23—22no—
393C23—22no—
394C23—24no—
395C23—22no—
396C23—24no—
397C23—22no—
398C23—42no—
399C23—42no—
400C23—48no—
401C23—42no—
402C23—42no—
403C23—84no—
404C23—81no—
405C23—81no—
406C23—1064no—
407C23—42416no—
408C23—4244no—
409C23—4244no—
410C23—2128no—
411C23—2124no—
412C23—2124no—
413C23—2124no—
414C23#1313318511yescharacteristic
415C23#1313318511yescharacteristic
416C23C2×#1171992412yes—
417C23C2×#1171992414yescomplemented
418C23#1313324212yes—
419C23C22×C22⋊D5311yescharacteristic
420Q8—28no—
421Q8—44no—
422Q8—48no—
423D4—4242no—
424D4—4244no—
425D4—2124no—
426D4—21216no—
427D4—2128no—
428D4—2124no—
429D4—2124no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—84no—
447C2×C4—1064no—
448C2×C4—2128no—
449C2×C4—2124no—
450C2×C4#1313318512yes—
451C2×C4C2×D4×D5312yes—
452C8—2124no—
453C22—24no—
454C22—24no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—24no—
460C22—44no—
461C22—48no—
462C22—42no—
463C22—42no—
464C22—82no—
465C22—84no—
466C22—42416no—
467C22—4248no—
468C22—2124no—
469C22—2124no—
470C22—2124no—
471C22#1681077312yes—
472C22#1681077312yes—
473C22#1681080012yes—
474C22#1681080912yes—
475C22C2×#1313318511yescharacteristic
476C22C22×#1171992411yescharacteristic
477C22C2×#1313324211yescharacteristic
478C4—22no—
479C4—22no—
480C4—42no—
481C4—44no—
482C4—82no—
483C4—2124no—
484C2—21no—
485C2—21no—
486C2—22no—
487C2—44no—
488C2—4244no—
489C2#2027366614yesdirect product factor
490C2C2×#1681077311yescharacteristic
491C2C2×#1681080011yescharacteristic
492C2C2×#1681080911yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#393C2
socleC23×C53#120
prime radicalC2×#16811185C2
hypercenterC2×#393D53
hypocenterC53C2×#1601
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16810762C2semidirect
3. #20273666C2direct (N × H)
4. C2×#16810356C2semidirect
5. C2×#16810347C2semidirect
6. C2×C53×#393C2semidirect
17. #16810762C22semidirect
19. C2×C22⋊#88322C22semidirect
20. #16810356C22semidirect
21. #16810347C22semidirect
22. C23×Q8×C53C22semidirect
26. C53×#393C22semidirect
28. C2×#95×C53C22semidirect
58. C22⋊#88322C23semidirect
64. C22×Q8×C53C23semidirect
77. #95×C53C23semidirect
83. C24×C53SD16semidirect
162. C23×C53C2×SD16semidirect
240. C2×#393D53semidirect
248. C2×C53#1601semidirect
253. C23×Q8C2×D53semidirect
270. #393C2×D53semidirect
272. C2×#95C2×D53semidirect
275. C53C2×#1601semidirect
282. C22×Q8C22×D53semidirect
324. #95C22×D53semidirect
335. C24#11719924semidirect
417. C23C2×#11719924semidirect
489. C2#20273666direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168111852
C2×#168107622
#202736662
C2×#168103562
C2×#168103472
C2×C53×#3932
C2×#160153